Suppose we have the self-adjoint Sturm-Liouville problem $$ (p(x)y')'+q(x)y=\lambda y \\ Ay(a)+By'(a)=0 \\ Ay(b)+By'(b)=0 $$ whose eigenvalues are $0<|\lambda_1|<|\lambda_2|<\cdots$ and $\{\phi_k(x)\}$ the corresponding eigenfunctions normalized so that their $L^2$-norm is 1. Supposing the sufficient regularity, or even in the case where $p$ and $q$ are constants, is there a way to bound $\|\phi_k\|_\infty$ in terms of $\lambda_k$? What about its derivatives, $\|\phi_k'\|_\infty$, $\|\phi_k''\|_\infty$, etc?
For a simple problem such as $y'' = \lambda y$ the boundedness can be checked by hand, and seems to be on the order of $\|\phi_k^{(n)}\|_\infty \leq C_{A, B} \ \lambda_k^{n/2}$.
Any insight or reference to a book where I can find a similar result or particular case would be greatly appreciated.